where in particular, the domain of integration simplifies from the random domain \(\hat{\mathcal {C}}_n\)  to the deterministic domain \(\mathcal {C}_n\) .
Actually from [1]}, we clearly don't need the strong condition that \(f\)  is analytic over all of \(. But this strong condition is easier to state and suffices for our paper.\)

This is actually a somewhat delicate argument and [2]} brushes through the finer argument details. First we note the complex integration surrounding Riesz functional calculus does not depend on the choice of curves.
